Mazur 13 port: exists_monic_square_root_of_relation

Proved

MazurProof.PadeZeroConstantSquare.exists_monic_square_root_of_relation

by xuanji · Oct 9, 2026 · Mathlib 0df444a (Lean v4.33.1)

huang-flt-portmazur-full-torsionmazur-n13-portorder13

A unit-times-square identity with a monic quadratic-or-smaller factor forces that factor to be a monic square.

This lemma is one step of a machine-checked proof that the genus-two curve

Y^2 = X^6+4X^5+6X^4+2X^3+X^2+2X+1

(a model of $X_1(13)$ ) has no rational affine points other than the cusps $X\in\{0,-1\}$ , which gives the case $N=13$ of Mazur's torsion theorem. The proof is Xiang Huang's Lean development, ported to this Mathlib and split into one node per large lemma; definitions live in the layered entries MazurN13_L0, MazurN13_L1, ….

Preamble

import Mathlib

set_option backward.isDefEq.respectTransparency false
set_option backward.isDefEq.respectTransparency.types false
open Polynomial
variable {K : Type*} [Field K]

Formal statement

theorem MazurProof.PadeZeroConstantSquare.exists_monic_square_root_of_relation {q a l u : K[X]} (hq : IsUnit q) (hl : l ≠ 0) (huMonic : u.Monic) (huDeg : u.natDegree ≤ 2) (hRelation : q * l ^ 2 = a ^ 2 * u) : ∃ b : K[X], b.Monic ∧ b.natDegree ≤ 1 ∧ u = b ^ 2 := by sorry

Source

Xiang Huang, FLT fork, https://github.com/xiangyazi24/FLT/blob/51bbb4f/FLT/Assumptions/MazurProof/PadeZeroConstantSquare.lean#L57

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